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REVIEW 5 major objections 5 minor 47 references

Rescaled Influence Functions: Accurate Data Attribution in High Dimension

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Rescaled influence functions replace each first-order influence estimate with a per-sample Newton step, keeping additivity and low cost while restoring accuracy when model dimension approaches or exceeds sample count.

desk verdict The empirical case for rescaled influence functions is real and worth engaging; the theory in Theorem 3.1 is, as written, unproven and the proof has a false lemma. read the letter →

arxiv 2506.06656 v2 pith:N7OSHYRE submitted 2025-06-07 cs.LG stat.ML

Ittai Rubinstein , Samuel B. Hopkins This is my paper
classification cs.LGstat.ML
keywords dataattributioninfluencefunctionsleave-one-outestimationNewtonstepapproximationhigh-dimensionalregressionoverparameterizedmodelspoisoningdetectionleveragescores
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that standard influence functions—first-order approximations used to predict how a model's parameters would change if training samples were removed—systematically underestimate those changes when the number of model parameters is comparable to or larger than the number of training samples. The authors propose the rescaled influence function (RIF), defined as the leave-one-out Newton step for each sample and added over the removed set, and show that in generalized linear models this is just a per-sample rescaling of the ordinary influence function by a leverage-score factor. Experiments on image, audio, and text classification show RIF tracking the true leave-T-out effect closely where IF underestimates, with negligible extra computation. A theorem bounds the gap between RIF and the full Newton-step estimate under cross-sample incoherence, explaining why RIF remains accurate in high dimension. The paper concludes by proposing RIF as a drop-in replacement for influence functions in data attribution applications.

What carries the argument

The rescaled influence function: the RIF for sample $i$ is the single Newton step for the leave-one-out loss, $\mathrm{RIF}_i = H^{-1}_{[n]\setminus\{i\}} \nabla\ell_i(x_i,y_i,\hat\theta)$, and RIF estimates for a removed set $T$ are the sum of $\mathrm{RIF}_i$ over $i \in T$. When each sample contributes a rank-one Hessian (generalized linear models), the leave-one-out Hessian inverse differs from $H^{-1}$ by a scalar, giving $\mathrm{RIF}_i = (1-h_i)^{-1} \mathrm{IF}_i$ with $h_i = \hat y_i(1-\hat y_i) x_i^\top H^{-1} x_i$ for logistic regression; this is the identity that lets RIF absorb the Hessian change at almost no extra cost. The proof of Theorem 3.1 is a matrix-perturbation analysis showing that the Newton-step inverse $H^{-1}_w$ is well approximated by keeping only single-sample Hessian corrections, with the error controlled by the cross-sample incoherence parameters $\delta$ and $\varepsilon$ from Assumption 3.

What would settle it

Construct a dataset with deliberately duplicated or near-collinear training samples so the incoherence parameters $\delta$ and $\varepsilon$ are large, then measure RIF's predicted leave-T-out effects against full retrains; if RIF remains accurate, the theoretical condition is not the operative explanation for the empirical success, whereas if RIF degrades, the condition names a concrete failure mode.

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Extended reading notes

Core claim

The central discovery is that the dominant error in influence functions comes from neglecting how removing a sample changes the Hessian, and that this error is repaired by an additive sum of leave-one-out Newton steps. For sample $i$, $\mathrm{RIF}_i = H^{-1}_{[n]\setminus\{i\}} \nabla \ell_i(x_i,y_i,\hat\theta)$, and for generalized linear models this equals $(1-h_i)^{-1} \mathrm{IF}_i$, where $h_i$ is a leverage score obtained from a single matrix-vector product with $H^{-1}$. Theorem 3.1 shows that under assumptions bounding each sample's Hessian share and cross-sample incoherence, the difference between RIF and the Newton-step estimate for removing up to $k$ samples is at most $k^2 \eta(1+2C_R)(\varepsilon + C_R C_\ell \delta)$. In a random-design least-squares heuristic, this scales as $O(k^2 \sqrt{d} \|\nabla f\|_2 / n^2)$, whereas the corresponding influence-function bounds from prior work scale as $O(k^2 d^4 \|\nabla f\|_2 / n^2)$; that improved dimension dependence is what keeps RIF useful when $n \approx d$ or $n \ll d$.

Load-bearing premise

The proof's error bound depends on the cross-sample incoherence assumption: the gradients and Hessians of any two distinct training samples must be nearly orthogonal when measured against the inverse total Hessian, and no single sample's Hessian may dominate the sum of the others; on data with heavy duplication or strong correlation this condition can fail and the bound can become vacuous.

Editorial extensions

If this is right

  • RIF can be substituted for IF in any smooth empirical-risk-minimization data-attribution pipeline while preserving additivity and most of the computational efficiency.
  • In high-dimensional and overparameterized regimes, RIF gives materially more accurate leave-T-out predictions than IF, with the improvement growing as $n/d$ shrinks or as $L2$ regularization weakens.
  • A simple data-poisoning attack that an IF-based detector misses is detected by RIF, supporting RIF as a better tool for robustness auditing.
  • The per-sample rescale factor bridges the known correlation-and-underestimation puzzle for influence functions: when most rescale factors agree, IF correlates with ground truth but lies below it.
  • The theoretical bound shows RIF tracks the Newton step for removal sets of size $k \ll n/\sqrt{d}$ in the random least-squares heuristic, a regime where the corresponding IF guarantee requires $k \ll n/d^2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The per-sample rescale factor $(1-h_i)^{-1}$ is itself a cheap diagnostic: a dataset where these factors spread widely is one where first-order attributions are least trustworthy.
  • Because the rescaling only corrects for Hessian change, RIF's edge over IF should shrink in strongly regularized or low-dimensional models; the paper's own experiments confirm this boundary, which is an implicit testable claim.
  • The same leave-one-out Hessian correction could be applied to other additive attribution tools, such as Shapley-value or datamodel estimators, wherever per-sample Hessians are low rank.
  • A practical extension would be to replace the total Hessian by a sketched or block-diagonal approximation in very large models, trading some of the correction for scalability; the paper notes Hessian inversion as a limitation but does not pursue this.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper introduces rescaled influence functions (RIF), an additive estimator of leave-T-out parameter effects formed by summing per-sample one-step Newton approximations instead of first-order influence-function vectors. The authors report experiments on logistic regression over image, text, and audio embedding datasets showing that RIF tracks ground-truth retraining behavior more closely than IF, present Theorem 3.1 as a theoretical explanation of this improvement, and demonstrate a simple poisoning-detection application. The manuscript also includes runtime comparisons, a discussion of related work, and an appendix with the proof of Theorem 3.1.

Significance. If the claims are sustained, RIF is a practically attractive drop-in replacement for influence functions: it preserves additivity, adds negligible overhead for generalized linear models, and appears to fix the systematic underestimation that has been observed for IF in high-dimensional problems. The paper's strengths include a reproducible experimental design with code, comparisons against full retraining rather than only proxy metrics, and a broad set of datasets. The main weakness is that the theoretical result is not established as written: the proof of Theorem 3.1 relies on Lemma A.2, which is false, and the application of the lemma does not match its hypotheses. The empirical section is visually suggestive but lacks reported quantitative accuracy metrics and error bars. These issues are serious but appear fixable, so the appropriate verdict is major revision.

major comments (5)
  1. [Appendix A, Lemma A.2] Lemma A.2 is false as stated. Take H = I and A_i = u_i u_i^T for three unit vectors in R^3 with pairwise inner products 0.1. Then sigma = 1 and delta_ij = 0.1 for all i != j, so the lemma's conclusion predicts norm <= 1 + sqrt(3)*0.1 approximately 1.173. The actual operator norm of sum_i u_i u_i^T is the largest eigenvalue of the Gram matrix, which is 1 + 2*0.1 = 1.2. Since the proof of Theorem 3.1 uses this lemma to bound ||G_w||_op and hence (I - G_w)^{-1}, the claimed bound on the Newton-step/RIF difference is not supported.
  2. [Appendix A, proof of Theorem 3.1] Even before Lemma A.2 is applied, the hypotheses are mismatched. The proof defines A_j = w_j H^{-1/2} H_j H^{-1/2}; for this choice the first hypothesis of Lemma A.2 requires control of ||H^{-1/2} A_j H^{-1/2}||_op = |w_j| ||H^{-1} H_j H^{-1}||_op, which is not what Assumption 2 bounds. The cross-term condition similarly requires ||sqrt(A_i) H^{-1} sqrt(A_j)||_op, which for these A_j does not reduce to ||H_i^{1/2} H^{-1} H_j^{1/2}||_op. Choosing A_j = w_j H_j would align the first hypothesis with Assumption 2, but the falsity of Lemma A.2 remains an independent blocking issue. The proof of Theorem 3.1 therefore needs a corrected matrix-concentration argument or a different route.
  3. [Section 2, Figures 1-2] The central empirical claim that RIF 'significantly outperforms' IF is presented only through scatter plots. No aggregate accuracy statistics, correlation coefficients, normalized errors, or confidence intervals are reported for the comparisons in Figures 1 and 2. Please add quantitative metrics, ideally per dataset and per removal fraction, with standard errors or bootstrap intervals, so that the magnitude and variability of the improvement can be assessed.
  4. [Section 3, Theorem 3.1 vs. experimental metrics] Theorem 3.1 bounds the inner product |<grad f(hat theta), hat theta_NS,w - hat theta_RIF,w>|, while the experiments in Figures 1 and 2 report differences in nonlinear function values such as test loss and self-loss. The passage from the inner-product bound to a bound on |f(hat theta_NS,w) - f(hat theta_RIF,w)| is not justified in the text; it would require additional control on the curvature of f along the segment between the two estimates. This connection should be made explicit, either as an additional assumption or as a stated limitation of the theory.
  5. [Section 3, Assumption 3] The theorem is vacuous unless delta and epsilon are small, since the condition k <= 1/(2 delta C_R) must hold. The paper does not report estimates of delta, epsilon, or C_R on any of the experimental datasets, so the theoretical result is not currently connected to the empirical setting in which RIF is claimed to help. Please either estimate these parameters empirically or explicitly state that the theorem is an idealized analysis that may not apply to the datasets in Section 2.
minor comments (5)
  1. [Appendix A, proof of Theorem 3.1] The proof refers to 'Assumption 5' when bounding ||grad f^T H^{-1} H_j^{1/2}||, but the manuscript states only Assumptions 1-4; this should be Assumption 4.
  2. [Section 5] The text contains the typo 'IMBD' in the Limitations paragraph; it should be 'IMDB'.
  3. [References] References [PGI+23a] and [PGI+23b] appear to be the same paper in published and arXiv form; consider citing only one to avoid duplication.
  4. [Section 5] The claim that the per-sample rescale factors are 'always larger than 1' should be qualified: the explicit formula 1/(1-h_i) applies in the logistic-regression/GLM setting described in Section 1.1, and for general low-rank Hessian contributions the relation is more involved and may require assumptions such as h_i < 1.
  5. [Figure 1] The captions and axis labels for Figures 1 and 2 contain the phrase '3 hidden' next to some columns; it is unclear what this refers to, and the figure captions would benefit from a clearer explanation of the layout.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: RIF is benchmarked against full retraining and its theory is a conditional bound from explicit assumptions, with no fitted target and no load-bearing self-citation.

full rationale

The paper's derivation chain has three independent legs. First, RIF is defined as the additive sum of single-sample Newton-step leave-one-out estimates, and the (1-h_i)^{-1} rescaling for generalized linear models is derived analytically from a low-rank Hessian update via Sherman-Morrison/Woodbury, not fitted to any target. Second, the empirical evaluation compares predicted leave-T-out effects against ground-truth full retraining, so the accuracy claims are externally benchmarked rather than self-confirming. Third, Theorem 3.1 is a conditional bound on the gap between the Newton-step and RIF estimates under explicitly stated Assumptions 1-4; its conclusion is a bound with stated constants, not an identity that re-imports the definition of RIF. Even if Appendix A's proof has an algebraic gap, as a skeptic contends, a flawed or incomplete proof is a correctness concern, not circularity. The only overlapping-author citation, [RH25], appears in a list of applications of influence functions to outlier and poisoning detection and is not load-bearing for the paper's central claims. The accuracy of Newton steps as a proxy for true leave-T-out effects is attributed to external prior work (Pregibon 1981, Koh et al. 2019), not to the authors' own prior results. No fitted input is renamed as a prediction, no ansatz is smuggled in via self-citation, and no known result is repackaged as new. I therefore find no significant circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The theoretical analysis rests on four quantitative assumptions about the loss geometry, plus a reliance on prior work for the accuracy of Newton steps. No new physical or conceptual entities are introduced, and the method has no free parameters fitted to the ground-truth effect.

free parameters (1)
  • L2 regularization strength lambda = 1e-5 for new datasets; values from prior work for DogFish and Enron
    Hand-chosen regularization in the experimental setup; it affects the comparison but is not fitted to the target result and is standard practice.
assumptions (5)
  • domain assumption Each per-sample loss is convex, so each Hessian H_i is positive semidefinite.
    Assumption 1 in Section 3; standard for logistic regression, but excludes non-convex neural network losses.
  • domain assumption No single sample has gradient or Hessian too large relative to the total Hessian: ||H^{-1/2} g_i||_2 <= C_L and ||H^{-1/2} H_i H^{-1/2}||_op <= 1 - 1/C_R.
    Assumption 2 in Section 3; key quantitative assumption that allows n ~ d, plausible for random data but not guaranteed.
  • domain assumption Cross-sample incoherence: for all i != j, ||H_i^{1/2} H^{-1} H_j^{1/2}||_op <= delta and ||H_i^{1/2} H^{-1} g_j||_2 <= epsilon.
    Assumption 3 in Section 3; the main structural assumption needed for the theorem, expected to hold in high dimension but not universally verified.
  • domain assumption Evaluation gradient projection control: ||H_i^{1/2} H^{-1} grad f(theta_hat)||_2 <= eta for all i.
    Assumption 4 in Section 3; bounds the interaction of the evaluation function with the per-sample geometry.
  • domain assumption The Newton step (NS) is an accurate approximation to the true leave-T-out effect, as established in prior work [KATL19].
    The theorem only bounds the RIF-NS gap; the paper relies on Koh et al. 2019 for the NS accuracy, which is cited but not proven here.

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Cite this review

Pith. "Pith review of Rescaled Influence Functions: Accurate Data Attribution in High Dimension." pith.science (2026). https://pith.science/paper/N7OSHYRE

@misc{pith2026250606656,
  author       = {Pith},
  title        = {Pith review of: Rescaled Influence Functions: Accurate Data Attribution in High Dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7OSHYRE}},
  note         = {Machine review of arXiv:2506.06656}
}
abstract

How does the training data affect a model's behavior? This is the question we seek to answer with data attribution. The leading practical approaches to data attribution are based on influence functions (IF). IFs utilize a first-order Taylor approximation to efficiently predict the effect of removing a set of samples from the training set without retraining the model, and are used in a wide variety of machine learning applications. However, especially in the high-dimensional regime (# params $\geq \Omega($# samples$)$), they are often imprecise and tend to underestimate the effect of sample removals, even for simple models such as logistic regression. We present rescaled influence functions (RIF), a new tool for data attribution which can be used as a drop-in replacement for influence functions, with little computational overhead but significant improvement in accuracy. We compare IF and RIF on a range of real-world datasets, showing that RIFs offer significantly better predictions in practice, and present a theoretical analysis explaining this improvement. Finally, we present a simple class of data poisoning attacks that would fool IF-based detections but would be detected by RIF.

Figures

Figures reproduced from arXiv: 2506.06656 by the authors.

Figure 1
Figure 1. Accuracy of IF versus RIF compared across datasets from image classification (DogFish, Cat [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. First row: accuracy of IF versus RIF compared across differing ratios of n and d, for the IMDB dataset, subsampled randomly to obtain datasets of varying sizes. IF and RIF are similar when n ≫ d, but as n decreases, RIF remains accurate while IF degrades. Second row: A similar comparison for the overparameterized DogFish dataset, where we vary the regularization strength λ. IF becomes accurate only under strong regu… view at source ↗
Figure 3
Figure 3. On the right we plot the actual vs predicted effect on a test samples logits from removing [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Accuracy of IF versus RIF compared across datasets from image classification (DogFish, [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.